Definition:Noetherian Ring

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Definition

Definition 1

A commutative ring with unity $A$ is Noetherian if and only if every ideal of $A$ is finitely generated.


Definition 2

A commutative ring with unity $A$ is Noetherian if and only if it satisfies the ascending chain condition on ideals.


Definition 3

A commutative ring with unity $A$ is Noetherian if and only if it satisfies the maximal condition on ideals.


Definition 4

A commutative ring with unity $A$ is Noetherian if and only if it is Noetherian as an $A$-module.


Also known as

Some sources render the term Noetherian ring as noetherian, dropping the capital N.


Also see

  • Results about Noetherian rings can be found here.


Source of Name

This entry was named for Emmy Noether.